1. Field
This patent specification relates to estimating properties from exponentially decaying data. More particularly, this patent specification relates to methods and systems for estimating material properties from a non-uniform sequence of excitations, such as with NMR measurements.
2. Background
Traditionally, fluid properties using Nuclear Magnetic Resonance (NMR) are obtained by study of longitudinal or transverse relaxation or diffusion measurements. Longitudinal (T1) relaxation data are acquired using an inversion-recovery pulse sequence where the recovery times are often chosen arbitrarily. Transverse (T2) relaxation data are acquired using the CPMG pulse sequence with pulses that are typically uniformly spaced in the time-domain. There is no clear mathematical methodology for choosing the acquisition times at which the diffusion measurements are obtained. The relaxation or diffusion data thus obtained from these pulse sequences are used to infer the relaxation time or diffusion distributions. In turn, these distributions are used to infer petrophysical or hydrocarbon properties.
Traditional well-logging using NMR is employed to infer petro-physical and fluid properties of the formation or in-situ hydrocarbon. The multi-exponential time-decay of NMR magnetization is characterized by relaxation time constants T1, T2 or diffusion D and corresponding amplitudes ƒT1(T1), ƒT2(T2) or ƒD(D). Since the relaxation times and/or diffusion constants are expected to be continuous and typically span several decades, these amplitudes are often referred to as relaxation-time or diffusion distributions. These distributions provide a wealth of information about both the rock as well as fluid properties. See e.g. D. Allen, S. Crary, and R. Freedman, How to use borehole Nuclear Magnetic Resonance, Oilfield Review, pages 34-57, 1997.
An NMR measurement consists of a sequence of transverse magnetic pulses transmitted by an antenna. Typically, the CPMG pulse sequence is used to measure T2 relaxation. It consists of a pulse that tips hydrogen protons 90° and is followed by several thousand pulses that refocus the protons by tipping them 180°. The data are acquired by an antenna between the evenly spaced 180° pulses and are thus uniformly spaced in time. On the other hand, the T1 relaxation is studied using the inversion-recovery pulse sequence. There is no specific rule for choosing the recovery times. See, Y. Q. Song, L. Venkataramanan, and L. Burcaw, Determining the resolution of Laplace inversion spectrum, Journal of Chemical Physics, page 104104, 2005. Similarly, there is no clear mathematical framework to optimally choose the acquisition times for diffusion measurements.
The data acquired from all of the above pulse sequences can be represented by a Fredholm integral (see, M. D. Hurlimann and L. Venkataramanan, Quantitative measurement of two dimensional distribution functions of diffusion and relaxation in grossly homogeneous fields, Journal Magnetic Resonance, 157:31-42, July 2002 and L. Venkataramanan, Y. Q. Song, and M. D. Hurlimann, Solving Fredholm integrals of the first kind with tensor product structure in 2 and 2.5 dimensions, IEEE Transactions on Signal Processing, 50:1017-1026, 2002, hereinafter “Venkataramanan et al. 2002”),
                              M          ⁡                      (            t            )                          =                              ∫            0            ∞                    ⁢                                    ⅇ                                                -                  t                                /                x                                      ⁢                                          f                x                            ⁡                              (                x                )                                      ⁢                                                  ⁢                          ⅆ              x                                                          (        1        )            
where the variable x typically refers to T1, T2, or 1/D. See, Venkataramanan et al. 2002. Thus the measured data in eqn. (1) is related to a Laplace transform of the unknown underlying function ƒx(x).
Scaling laws established on mixtures of alkanes provide a relation between fluid composition and relaxation time and/or diffusion coefficients of the components of the mixture (see D. E. Freed, Scaling laws for diffusion coefficients in mixtures of alkanes, Physical Review Letters, 94:067602, 2005, and D. E. Freed, Dependence on chain length of NMR relaxation times in mixtures of alkanes, Journal of Chemical Physics, 126:174502, 2007). A typical crude oil has a number of components, such as methane, ethane etc. Let Ni refer to the number of carbon atoms or chain length of the i-th component. Let N denote the molar average chain length of the crude oil, also defined as the harmonic mean of the chain lengths,
                                          1                          N              _                                ≡                                    ∑                              i                =                1                            ∞                        ⁢                                                  ⁢                                                            f                  N                                ⁡                                  (                                      N                    i                                    )                                                            N                i                                                    =                  〈                      N                          -              1                                〉                                    (        2        )            where ƒN(Ni) denotes the mass fraction of component i in the hydrocarbon. According to the scaling law, the bulk relaxation time T2,i of component i in a crude oil follows a power law and depends inversely on its chain length or number of carbon atoms, Ni. Thus, the larger the molecule, the smaller the relaxation time T2,i. The relaxation time also depends inversely on the average chain length N of the hydrocarbon, according toT2,i=BNi−κN−γ  (3)
In eqn. (3), parameters B, γ and κ are constants at a given pressure and temperature. Similarly, if Di refers to the diffusion coefficient of component i,Di=ANi−νN−β  (4)where A, ν and β and are constants at a given pressure and temperature. The scaling theory can also be used to derive an expression for fluid viscosity,η∝D−1η∝NβNν.  (5)
Traditional analysis of the measured NMR data to obtain fluid properties goes as follows. The inverse Laplace transform of eqn. (1) is performed to estimate the probability density function ƒx(x) from the measured data. Next, using eqns. (3)-(5), the average chain length N and the chain length distribution ƒN(N) are computed. Viscosity η of the hydrocarbon is then estimated using eqn. (5). This traditional analysis has some disadvantages. First, the inverse Laplace transform of the measured NMR data in eqn. (1) is non-linear due to the non-negativity constraint on ƒx(x). See, Venkataramanan et al. 2002, and E. J. Fordham, A. Sezginer, and L. D. Hall, Imaging multiexponential relaxation in the (y, loge T1) plane, with application to clay filtration in rock cores, J. Mag. Resonance A, 113:139-150, 1995. Next, it is also mathematically ill-conditioned. Often, regularization or prior information about the expected solution is incorporated into the problem formulation to make it better conditioned. However, the choice of the regularization functional as well as the weight given to the prior information is a well-known drawback of this transform due to its non-uniqueness. See, W. H. Press, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C, Cambridge University Press, 1992. Thus errors in the estimation of T1, T2, or D distributions are propagated into errors in the estimation of fluid properties.